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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 157532, 13 pages
http://dx.doi.org/10.1155/2012/157532
Research Article

α-Well-Posedness for Quasivariational Inequality Problems

School of Mathematics, Chongqing Normal University, Chongqing 400047, China

Received 21 February 2012; Accepted 18 March 2012

Academic Editor: Simeon Reich

Copyright © 2012 Jian Wen Peng and Jing Tang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce and study the concepts of α-well-posedness and L-α-well-posedness for quasivariational inequality problems having a unique solution and the concepts of α-well-posedness in the generalized sense and L-α-well-posedness in the generalized sense for quasivariational inequality problems having more than one solution. We present some necessary and/or sufficient conditions for the various kinds of well-posedness to occur. Our results generalize and strengthen previously known results for quasivariational inequality problems.

1. Introduction

Let be a reflexive real Banach space and let be a nonempty closed convex subset of . Let be a set-valued mapping from to and let be an operator from to the dual space . Bensoussan and Lions [1], Baiocchi and Capelo [2], and Mosco [3] considered the following quasivariational inequality (in short, (QVIP)), which is to find a point such that

The interest in quasivariational inequality problems lies in the fact that many economic or engineering problems are modeled through them, as explained in [4, 5] where a wide bibliography on variational inequalities, quasivariational inequality problems, and related problems is contained. Moreover, under suitable assumptions, a quasivariational inequality is equivalent to a generalized Nash equilibrium problem [3].

On the other hand, well-posedness plays a crucial role in the stability theory for optimization problems, which guarantees that, for an approximating solution sequence, there exists a subsequence which converges to a solution [6]. The study of well-posedness for scalar minimization problems started from Tikhonov [7] and Levitin and Polyak [8]. Since the convergence of numerical methods for quasivariational inequality Problems can be obtained also with the aid of well-posedness theory, Lignola [9] introduced and investigated the concepts of well-posedness and L-well-posedness for quasivariational inequalities having a unique solution and the concepts of well-posedness and L-well-posedness in the generalized sense for quasivariational inequality problems having more than one solution.

In this paper, inspired by the above concepts of well-posedness for (QVIP), we introduce and study the concepts of -well-posedness and L--well-posedness for quasivariational inequality Problems having a unique solution and the concepts of -well-posedness in the generalized sense and L--well-posedness in the generalized sense for quasivariational inequality Problems having more than one solution. The results in this paper generalize and improve the results in [9, 10].

2. Preliminaries

Denote by the solution set of (QVIP). Let . In order to investigate the -well-posed for (QVIP), we need the following definitions.

First we recall the notion of Mosco convergence [11]. A sequence of subsets of Mosco converges to a set if where and are, respectively, the Painlevé-Kuratowski strong limit inferior and weak limit superior of a sequence , that is, where “” means weak convergence, “” means strong convergence.

If , we call the sequence of subsets of Lower Semi-Mosco which converges to a set .

It is easy to see that a sequence of subsets of Mosco converges to a set which implies that the sequence , also Lower Semi-Mosco, converges to the set , but the converse is not true in general.

We will use the usual abbreviations usc and lsc for “upper semicontinuous” and “lower semicontinuous,” respectively. Let be a reflexive real Banach space with dual . An operator will be called hemicontinuous if it is continuous from every segment of to endowed with the weak topology. will be called monotone if for every , and . will be called pseudomonotone if for every and .

Definition 2.1. A sequence is an -approximating sequence for (QVIP) if
(i), for all ;(ii)there exists a sequence , decreasing to 0 such that

Definition 2.2. A quasivariational inequality (QVIP) is said to be -well-posed (resp., -well-posed in the generalized sense) if it has a unique solution and every -approximating sequence strongly converges to (resp., if the solution set of (QVIP) is nonempty and for every -approximating sequence has a subsequence which strongly converges to a point of ).

Definition 2.3. A sequence is an L--approximating sequence for (QVIP) if:
(i), for all ;(ii)there exists a sequence , decreasing to 0 such that , and

Definition 2.4. A quasivariational inequality (QVIP) is said to be L--well-posed (resp., L--well-posed in the generalized sense) if it has a unique solution and every L--approximating sequence strongly converges to (resp., if the solution set of (QVIP) is nonempty and for every L--approximating sequence has a subsequence which strongly converges to a point of ).
It is worth noting that if , then the definitions of -well-posedness, -well-posedness in the generalized sense, L--well-posedness, and L--well-posedness in the generalized sense for (QVIP), respectively, reduce to those of the well-posedness, well-posedness in the generalized sense, L-well-posedness, and L-well-posedness in the generalized sense for (QVIP) in [9]. We also note that Definition 2.2 generalizes and extends -well-posedness and -well-posedness in the generalized sense of variational inequalities in [10] which are related to the continuously differentiable gap function of variational inequality Problems introduced by Fukushima [12].

We recall some lemmas which will be needed in the rest of this paper.

Lemma 2.5 (see [13]). Let be a sequence of nonempty subsets of the space such that
(i) is convex for every ;(ii);(iii)there exists such that .
Then, for every , there exists a positive real number such that , for all .
If is a finite dimensional space, then assumption (iii) can be replaced by
(iii)’.

The following Lemmas 2.6 and 2.7 play important roles in this paper. Now we present a Minty type lemma for quasivariational inequalities as follows.

Lemma 2.6. Suppose that set-valued mapping is nonempty convex-valued, the operator is hemicontinuous and monotone, . Then the following conditions are equivalent:
(i), for all ,(ii), for all .

Proof. We first prove that (ii) implies (i). Let be a arbitrary point of . For every number , since the set-valued mapping is convex-valued and , the point belongs to . It follows from (ii) that From the definition of , one has and it follows from the hemicontinuity of that then The converse is an easy consequence of monotonicity of .

Lemma 2.7. Assume that set-valued mapping is nonempty convex-valued, then if and only if the following conditions hold:

Proof. The necessity is clearly held. Now we prove the sufficiency. Let for all , for all , . Since is convex-valued, , one has which implies that The above inequality implies, for converging to zero, that is a solution of (QVIP). This completes the proof.

3. Case of a Unique Solution

In this section, we investigate some metric characterizations of -well-posedness and --well-posedness for (QVIP).

For any , we consider the set

Theorem 3.1. Let the same assumptions be as in Lemma 2.7. Then, one has(a)(QVIP) is -well-posed if and only if the solution set of (QVIP) is nonempty and ;(b)moreover, if is pseudomonotone, then (QVIP) is --well-posed if and only if the solution set of (QVIP) is nonempty and .

Proof. We only prove (a). The proof of (b) is similar and is omitted here. Suppose that (QVIP) is -well-posed, then . It follows from Lemma 2.7 that . Suppose by contradiction that there exists a real number , such that , then there exists , with , and , such that , for all . Since the sequences are both -approximating sequences for (QVIP), and strongly converge to the unique solution , and this gives a contradiction. Therefore, .
Conversely, let , be an -approximating sequence for (QVIP). Then there exists a sequence , with , such that that is, , for all . It is easy to see and implying that is a singleton point set. Indeed, if there exist two different solutions , then from Lemma 2.7, we know that , for all . Thus, , a contraction. Let be the unique solution of (QVIP). It follows from Lemma 2.7 that . Thus, . So strongly converge to . Therefore, (QVIP) is -well-posed.

Theorem 3.2. Let and the following assumptions hold:
(i)the set-valued mapping is nonempty convex-valued, and, for each sequence in converges to , the sequence Lower Semi-Mosco converging to ;(ii)for every converging sequence , there exists , such that(iii)the operator is hemicontinuous and monotone on .Then, (QVIP) is -well-posed if and only if

Proof. The necessity has been proved in Theorem 3.1(a).
Conversely, assume that (3.4) holds. It is easy to see that (3.4) implies that the solution set of (QVIP) is a singleton point set. Let be an -approximating sequence for (QVIP), that is, there exists a sequence , with , such that Therefore, , for all . In light of (3.4), is a Cauchy sequence and strongly converges to a point . In order to obtain that solves (QVIP), we start to prove that . For each , choose , such that . It follows from and that . It follows from the assumption (i) that . Thus, .
To complete the proof, consider an arbitrary point . By Lower Semi-Mosco convergence again, we have . Also observe that assumption (ii) applied to the constant sequence , for all , implies that . From Lemma 2.5, it follows that if , then there exist and such that , for all . Thus, for sufficiently large. Notice the is monotone and the sequence is an -approximating sequence for (QVIP), then we have If , let be a sequence converging to , whose point belongs to a segment contained in . Since , for all , one has Since the hemicontinuity of , It follows from Lemma 2.6 that then, by Lemma 2.7, we obtain that solves (QVIP). This completes the proof.

Now, we present a result in which assumption (ii) of above theorem is dropped, while the continuity assumption on the operator is strengthened.

Theorem 3.3. Let the following assumptions hold:(i)the set-valued mapping is nonempty convex-valued, and, for each sequence in converging to , the sequence Lower Semi-Mosco converges to ;(ii)the operator is -continuous on .
Then, (QVIP) is -well-posed if and only if (3.4) holds.

Proof. The necessity follows from Theorem 3.1 and Lemma 2.7.
Conversely, let be an -approximating sequence for (QVIP) and (3.4) holds. From (3.4) and the proof of Theorem 3.2, we can obtain that strongly converges to , with . Since Lower Semi-Mosco convergence implies for every , there exists sequence strongly converging to such that . Since the operator is -continuous and is an -approximating sequence for (QVIP), we have By Lemma 2.7, we obtain that solves (QVIP). This completes the proof.

Theorem 3.4. Let the following assumptions hold:
(i)the set-valued mapping is nonempty convex-valued, and, for each sequence in converges to , the sequence Lower Semi-Mosco converging to ;(ii)for every converging sequence , there exists , such that(iii)the operator is hemicontinuous and monotone on .
Then, (QVIP) is L--well-posed if and only if

Proof. Assume that (QVIP) is L--well-posed, then it follows from the monotonicity of that , for all . It follows from Theorem 3.1(b) that the necessity can be completed.
Assume that (3.12) holds. Let be an L--approximating sequence for (QVIP), then there exists a sequence , with , such that , for all . Following the same argument as the proof of Theorem 3.1, it is easy to see and imply that is a singleton point set. In light of the assumption, is a Cauchy sequence and strongly converges to a point and . Let and using Lemma 2.5, one has , for sufficiently large. Then, we get If , let a sequence converges to , whose points belong to a segment contained in . Since and the operator is hemicontinuous, one gets According to Lemmas 2.6 and 2.7, is the solution of (QVIP).

Theorem 3.5. Let the following assumptions hold:(i)the set-valued mapping is nonempty convex-valued, and, for each sequence in converging to , the sequence Lower Semi-Mosco converges to ;(ii)the operator is -continuous and monotone on .
Then, (QVIP) is L--well-posed if and only if (3.12) holds.

Proof. Assume (3.12) holds. Let be an L--approximating sequence for (QVIP), then there exists a sequence , with , such that , for all . Since , is a Cauchy sequence and converges to . As the similar proof of Theorem 3.2, . Let . Since Lower Semi-Mosco convergence implies for every , there exists a sequence converging to , such that . Since is -continuous and is an L--approximating sequence for (QVIP), one has Applying Lemmas 2.6 and 2.7, we have that (QVIP) is L--well-posed.
The necessity can be completed as Theorem 3.3.

4. -Well-Posedness in the Generalized Sense

In this section, we introduce and investigate some metric characterizations of -well-posedness in the generalized sense and --well-posedness in the generalized sense for (QVI).

Definition 4.1 (see [11]). Let be a metric space and let be nonempty subsets of . The Hausdorff distance between and is defined by where with .

Definition 4.2 (see [11]). Let be a nonempty subset of . The measure of non compactness of the set is defined by where diam means the diameter of a set.

Theorem 4.3. Let the same assumptions be as in Lemma 2.7. Then, one has the following.(a)(QVIP) is -well-posed in the generalized sense if and only if the solution set of (QVIP) is nonempty compact and .(b)Moreover, if is pseudomonotone, then (QVIP) is --well-posed in the generalized sense if and only if the solution set of (QVIP) is nonempty compact and .

Proof. We only prove (a), the proof of (b) is similar and is omitted here. Assume that (QVIP) is -well-posed in the generalized sense, then the is nonempty and compact. It follows from Lemma 2.7 that . Now we prove . Suppose by contradiction that there exists , and , such that . It follows from that is an -approximating sequence for (QVIP). (QVIP) is -well-posedness in the generalized sense, then there exists a subsequence of strongly converging to a point of . This contradicts . Thus, .
For the converse, let be an -approximating sequence for (QVIP), then . It follows from that there exists a sequence , such that . Since is compact, there exists a subsequence of strongly converging to . Thus there exists the corresponding subsequence of strongly converging to . Therefore, (QVIP) is -well-posed in the generalized sense.

Theorem 4.4. (a) If (QVIP) is -well-posed in the generalized sense, then
(b) If (4.3) and the following assumptions hold:
(i)the set-valued mapping is nonempty convex-valued, and, for each sequence in converges to , the sequence Lower Semi-Mosco converging to ;(ii)the operator is -continuous on ,then, (QVIP) is -well-posed in the generalized sense.

Proof. (a) Suppose that (QVIP) is -well-posed in the generalized sense. So , for all . By Theorem 4.3(a), is nonempty compact and . For any , we have and since is compact, . For every , the following relation holds [14]: It follows from , that .
(b) Assume that (4.3) holds. Then, for any , cl is nonempty closed and increasing with . By (4.3), , where cl is the closure of . By the generalized Cantor theorem [11, page 412], we know that where is nonempty compact.
Now we show that It follows from Lemma 2.7 that . So we need to prove that . Indeed, let . Then, for every . Given , , for every , there exists such that . Hence, and It follows from (4.8), , and the proof of Theorem 3.2 that .
Since Lower Semi-Mosco convergence implies that, for every , there exists a sequence , for all , such that in the strongly topology.
Since the operator is -continuous on , hence By Lemma 2.7, we know . Thus, . It follows from (4.6) and (4.7) that . It follows from the compactness of and Theorem 4.3(a) that (QVIP) is -well-posed in the generalized sense. The proof is completed.

Theorem 4.5. Let be a nonempty, compact, and convex subset of , let the set-valued mapping be nonempty convex-valued, and, for each sequence in converging to , the sequence Lower Semi-Mosco converges to , and the operator is -continuous on . Then, (QVIP) is -well-posed in the generalized sense.

Proof. Let be an -approximating sequence for (QVIP). Since the set is compact, there exists subsequence of strongly converging to a point . Reasoning as in Theorem 3.3, we get and solves (QVIP). Therefore, (QVIP) is -well-posed in the generalized sense.

Theorem 4.6. Let the following assumptions hold:
(i)the set-valued mapping is nonempty convex-valued, and, for each sequence in converging to , the sequence Lower Semi-Mosco converges to ;(ii)the operator is -continuous and monotone on .
Then, (QVIP) is L--well-posed in the generalized sense if and only if

Proof. Assume that (QVIP) is L--well-posed in the generalized sense. It follows from Lemma 2.7 and the monotonicity of that , for all . And so , for each . By Theorem 4.3(b), we can get as . From the proof of Theorem 4.4, we also obtain Thus, .
Conversely, assume (4.11) holds. Then, for any , cl is nonempty closed and increasing with . By (4.11), , where cl is the closure of . By the generalized Cantor theorem [11, page 412], we know that where is nonempty compact.
Now we show that It follow from Lemma 2.7 and the monotonicity of that . So we need to prove that . Indeed, let . Then for every . Given , , for every , there exists such that . Hence, and
It follows from (4.15), , and the proof of Theorem 3.2 that .
Since Lower Semi-Mosco converges to , for every , there exists a sequence , for all , such that in the strong topology.
Since the operator is -continuous on , hence By Lemma 2.6 we know that , such that It follow from Lemma 2.7 that . Thus, . It follows from (4.13) and (4.14) that . It follows from the compactness of and Theorem 4.3(b) that (QVIP) is L--well-posed in the generalized sense. The problem is completed.

Remark 4.7. It is easy to see that if , then by the main results in our paper, we can recover the corresponding results in [9] with the weaker condition that Lower Semi-Mosco converges to instead of the condition that is -closed and -subcontinuous, and -lower semicontinuous.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant no. 11171363 and Grant no. 10831009), the Natural Science Foundation of Chongqing (Grant no. CSTC, 2009BB8240), the Special Fund of Chongqing Key Laboratory (CSTC 2011KLORSE01) and the Project of the Third Batch Support Program for Excellent Talents of Chongqing City High Colleges.

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